4 Answers
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The use of flash actually pre-dates electric lighting, the first flashes being entirely chemical.

Nowadays the main reason for using flash is that a flash can be considerably brighter than continuous lighting because it only has to be on for a very short time (while the photo is being taken).

A continuous light source that was the same brightness as a flash would consume a huge amount of energy (and get exceedingly hot during the process). Fluorescent lighting has come on a long way and produces much less heat but at high power output can still cook your subject. As it stands a simple hotshoe flash can produce many many times the light intensity of a mains powered bulb, despite running on AA batteries because it is only on for one thousandth of a second. The flash works by charging a capacitor slowly and releasing all the energy in a single burst.

Other than the greater quantity of light (which enables you to use a lower ISO, and to light a larger area) there are other advantages to flashes:

The short duration allows you to freeze motion as any subject motion after the short burst of light will not be recorded.

The shutter speed independent nature of flash allows you to tune the balance between flash and ambient light. Aperture/ISO have an effect on both flash and ambient brightness, shutter speed only effects ambient brightness, by changing shutter speed an compensating with aperture you can alter the flash/ambient light balance.

We don't, necessarily. As Matt pointed out, there are some significant advantages to flash, particularly when it is being mixed with existing lighting. But whereas flash used to be the only sane way to light in a studio setting (since tungsten lights are extremely hot, HMI lights are ridiculously expensive, and films with decent grain and early digital camera usable ISO ranges required boatloads of light), other lights can be a reasonable alternative for many purposes these days.

Daylight-balanced fluorescents with good colour rendering and high-frequency ballasts, paired with digital cameras having excellent noise/grain characteristics at higher ISOs make constant light a reasonable alternative. It would have been nearly unthinkable to use a 400-speed colour film for portraiture on a small-format (35mm) camera, and a bit risky on medium format (since it would have really limited the enlargement). ISO 400 is just another setting on any reasonably modern DSLR or mirrorless interchangeable lens camera. Most produce eaasily cleaned-up images at ISO 1600 (in 35mm film terms, that's gritty "mean streets" black-and white photojournalism), and some high-end cameras are perfectly happy at ISO 3200 or 6400 (surveillance quality at best in the old days).

You can do professional-quality work with some CFL lamps, either on a tight budget using DIY light modifiers or by purchasing lights aimed at professionals (like the Westcott Spiderlite or the Lowel Rifa). The characteristic green cast and flicker of fluorescents is largely restricted to the typical 40-watt tubes and industrial lighting; most daylight-balanced CFLs have electronic (rather than magnetic) ballasts and decent, correctable colour. Bulbs made for photography tend to have very good spectra, high output, and flicker frequencies that make shutter speeds in the 1/250th second region or longer practical.

Long-tube lights are also available, but it's harder to find fixtures with high-frequency ballasts (to avoid flicker) without spending serious money (like on a Kino Flo setup, which has the advantage of being nearly infinitely adjustable for brightness). The main drawback is that they're not really dimmable unless the ballast frequency is very high, since dimming a fluorescent (or an LED) is really a matter of adjusting the amount of time they're on and off, so turning bulbs on or off in a multi-bulb fixture or moving the lights are the only adjustments you get at the cheap end. (Tungsten bulbs are dimmable, but they change colour when you dim them.)

LEDs are pretty good as well these days, but they're horrendously expensive at the moment unless you're shooting still lives (where slow shutter speeds don't matter). In fact, the "flash" on camera phones is just a high-powered LED that would suck the battery dry in no time if it stayed on.

Constant lighting has the advantage of letting you see the light exactly as it is before you take the picture. Your eyes might fib a little about the contrast, but you'll be able to see exactly where the light and shadow are. With portable flash, you have to guess, and even with studio strobes having modelling lights, it's just an approximation—even with the frosted covers on units like the Paul C. Buff Einstein, Profoto and the Elinchrom Style series, the modelling light isn't identical to the flash.

It's the usable high ISOs that make these continuous lights practical. As long as you don't want to freeze action, you can keep the light level down to something tolerable (even if it seems really, really bright by typical home or office lighting standards). If you were shooting in the range of ISO 25-100, the light would have to be blindingly bright, and you'd still need to use relatively low shutter speeds. An 85-watt CFL in a softbox two feet from your subject's face might seem like a lot, but imagine the experience with a 2000-watt hot halogen light at the same distance. (We would have probably used a 10-12KW HMI further back in the old days to solve the heat problem, but then you lose the fall-off. And that still wouldn't make it feel any less like somebody stuck a 2000-watt light bulb in your face.) We can use less light from much cooler sources and get high-quality results. Flash is still irreplaceable for a lot of applications, but it's not the only solution.

At the speeds that a camera works, "simple light bulbs" don't work well at all. If its a traditional tungsten bulb (in use for 100+ years) then the lamp generates far more heat than light, and the light is red/yellow in color. As @matt said, to get the same amount of light would take a huge lamp that would draw huge amounts of power (think multiple toasters).

If its a new compact fluorescent bulb, the light is not only green, but it flickers, going off and on at a frequency of 120 hz. Which means that if you try to shoot faster than 1/125, you can't be sure whether the lamp will be on or off.

We use flash because we take photographs with light. If there is not enough light, we need something to add more light. Strobes work well for their engineering requirements.

Well, i speak now only as a naturephotographer. I dont ever use flash. Well, i did try that and worked with it, but came up with the solution that no matter what it is, its all about the nature light, sun and such. I know you have to use flash in several situations, but as far as it goes with me - i gave up using it years ago. Best lights are early in the morning and late in the evening.

One time we were at one national park, taking photos and it was getting pretty dark. Most of shooters said that "ok, time to leave", but then - many started to take photos. With eyes you didnt see much of light, but still, then we were taking those photos. And wow, the light was just perfect when saw the images via computer.

So, the point of this is, that it really depends. And when it comes to taking photos in wild, dont believe your eyes when it comes to light and such....